# Course:Study Guide - Exam 1 - Bart-Fall08

## Part I: Essay type questions

The exam is 50 minutes long.

You should be thorough in your answer. Less than a half a page is likely not detailed enough of an explanation. I will ask some or all of the following questions:

**Compare and contrast the rozette, the frieze and the wallpaper symmetry groups**

What are we looking for when we determine these symmetry groups? What types of symmetries do we look for in a rozette? In a frieze pattern? In a wallpaper pattern? I.e which patterns have reflections? Which ones have rotations? Of what degree? Which ones can have translations? In what directions? Anything else worth mentioning? Provide a (simple) example of each to show the difference. (Be thorough!)

**Show carefully that all parallelograms tessellate.**

Be thorough in your explanations! How do you argue you can cover the plane without gaps and overlaps? What role do the angle measures play?

**Show carefully that all triangles tessellate.**

Be thorough in your explanations! What role do the angle measures play? Be detailed in your explanation.

**Why are there exactly 3 regular tessellations?**

First explain what a regular tessellation is. Then show carefully that there can only be three. Use the outline of the explorations we used. Give as much detail as possible.

**How does one in general use geometry to create Escher-like tessellations?**

What kind of underlying geometric tessellations do we use? Does it matter which geometric tessellation we start with? For instance: if we want to create a tessellation with rotational symmetry, are we somewhat forced to start with a given geometric tessellation? If so, which one(s)? How do we create the tessellations by recognizable figures?

## Part II: Problems based on explorations and homework

- I will take some questions from the homework.
- I will take some questions from the explorations. See above for a list of all the explorations we have covered in class.